Categoricity and Consistency in Second-Order Logic

We analyse the concept of a second-order characterisable structure and divide this concept into two parts—consistency and categoricity—with different strength and nature. We argue that categorical characterisation of mathematical structures in second-order logic is meaningful and possible without assuming that the semantics of second-order logic is defined in set theory. This extends also to the so-called Henkin structures.     

Profiles of Contextual Risk at Birth and Adolescent Substance Use

This study examined whether there are subgroups of families with distinct profiles of prenatal/birth contextual risk, and whether subgroup membership was differentially related to adolescent substance use. Data from the Northern Finland Birth Cohort 1986 were used. A five-class model provided the most meaningful solution. Large Family Size (7.72%) and Low Risk (69.69%) groups had the lowest levels of alcohol, cigarette, and illegal drug use. Similar high levels for each of the three substance-related outcomes were found for Parent Substance Misuse (11.20%), Maternal School Dropout (4.66%), and Socioeconomic Disadvantage (6.72%) groups. Maternal smoking and drinking while pregnant and paternal heavy alcohol use were found to be key prenatal risk factors that tended to cluster together and co-occur with other prenatal risk factors differently for different subgroups of youth.     

On the Expressive Power of Monotone Natural Language Quantifiers over Finite Models

We study definability in terms of monotone generalized quantifiers satisfying Isomorphism Closure, Conservativity and Extension. Among the quantifiers with the latter three properties – here called CE quantifiers – one finds the interpretations of determiner phrases in natural languages. The property of monotonicity is also linguistically ubiquitous, though some determiners like an even number of are highly non-monotone. They are nevertheless definable in terms of monotone CE quantifiers: we give a necessary and sufficient condition for such definability. We further identify a stronger form of monotonicity, called smoothness, which also has linguistic relevance, and we extend our considerations to smooth quantifiers. The results lead us to propose two tentative universals concerning monotonicity and natural language quantification. The notions involved as well as our proofs are presented using a graphical representation of quantifiers in the so-called number triangle.